Questions tagged [projective-modules]

For questions about projective modules over a ring and projective objects in related categories.

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Is there a standard reference for: taking projective covers of simple modules commutes with finite Galois field extensions?

Let $\Lambda$ be an Artin algebra over a finite field $k$. Is there a standard reference for the fact that taking projective covers of simple $\Lambda$-modules commutes with finite Galois field ...
0 votes
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162 views

The coevaluation map for a projective module and its dual

$\DeclareMathOperator\coev{coev}$Let $R$ be a noncommutative ring and let $P$ be a bimodule over $R$, that is finitely generated and projective as a left module. It is "well-known" that any ...
1 vote
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Prove that $f(M)=f^2(M)$ implies $f(M)$ is a direct summand of $M$ whenever $\text{End}_R(M)$ is a reduced ring

Let $M$ be a right $R$-module with the property that every homomorphism $\gamma:Sf\to M, f\in S=\text{End}_R(M)$, extends to $S\to M$. If $S$ has the property $f^2=0$ implies $f=0$ for every $f\in S$...
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1 answer
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Example of non vanishing Ext

Let $R$ be a commutative Noetherian ring and $I$ is a proper ideal of $R$. suppose that $M$ is a f.g. $R$-module. $\DeclareMathOperator\Ext{Ext}$I'm looking for an example that has this property: $$\...
1 vote
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Existence of a finite resolution

I have tried to formulate a question in which I was very curious, any hints suggestions are also welcomed. Thanks in advance. Let $M$ be an $R$ module ($R$ commutative ring with unity). It is given ...
1 vote
0 answers
81 views

Composition of faithfully flat ring extensions

Let $R$ be a not necessarily commutative, unital, ring, and for simplicity let module always mean right module. We say that a unital ring extension $R \hookrightarrow S$ is flat, or faithfully flat, ...
3 votes
1 answer
184 views

Faithful flatness for rings

Let $R$ be a ring and let $M$ be a right module over $R$. We say that $M$ is faithfully flat as a right module if the functor $M \otimes_R -$ from left $R$-modules to abelian groups that preserves ...
4 votes
1 answer
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In a category with a projective generator, do morphisms from the generator determine the object?

I have a cocomplete abelian category $\mathcal C$ and two objects $X$, $Y$ in $\mathcal C$. Further, $\mathcal C$ has a projective generator $P$. I have an isomorphism $$ \mathcal C(P,X) \cong \...
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2 votes
2 answers
180 views

An algebra which is a direct sum of simple sub-bimodules over a subalgebra

Let $A$ be an infinite-dimensional noncommutative algebra over a field, let $B$ be an infinite-dimensional subalgebra of $A$, and let $A$ be a direct sum of projective simple $B$-sub-bimodules. Then ...
4 votes
1 answer
197 views

Are finite projective modules over $R[t]$ free when $R$ is DVR?

Let $R$ be a discrete valuation ring (DVR) and let $M$ be a projective module of finite type over the polynomial ring $R[t]$. Is $M$ free over $R[t]$? As far as I understand, this should be a ...
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Profinite projective and free modules

I am studying cohomology of profinite groups and the following question came to my mind: suppose we have $G$ a pro-$p$ group which is Poincaré Dual of dimension $d$. This means that $\mathbb{Z}_p$ as ...
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4 votes
1 answer
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Are finitely presented algebras over VNRs projective?

Question: Let $A$ be a commutative von Neumann regular ring, and $B$ an $A$-algebra of finite presentation, i.e. $B = A[x_1, \ldots, x_n]/(f_1, \ldots, f_m)$. Is $B$ a projective $A$-module? In the ...
4 votes
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Rings such that torsion-free/flat/projective modules are flat/projective/free

While thinking about this question (and specifically YCor's remarks), I tried to remember what can be said about rings such that every torsion-free module is free, and I could not; and such things, ...
2 votes
2 answers
594 views

Characterization of projective modules in terms of Ext groups

This is from Hartshrone exercise 6.6 part (a). Let $A$ be a regular local ring and $M$ be a finitely generated $A$-module, prove the following $M$ is projective $\iff$ $\operatorname{Ext}^{i}(M,A)=\{...
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2 answers
284 views

Projective modules restricted to smooth curves

I asked this question on Stack Exchange, but no one answered this. I want to prove a coherent sheaf $M$ on $X$ is locally free if and only if this is true for $M|_{X'}$ , for all smooth curves $X'$ ...
3 votes
1 answer
171 views

Is a tower of locally-free modules locally a tower of free modules?

Suppose we have a (commutative, unital) ring $R$ and a (commutative, unital) $R$-algebra $A$ such that $A$ is projective of constant rank $n$ as an $R$-module. This condition is equivalent to there ...
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8 votes
2 answers
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tangent bundle on noncommutative manifold

Using the Serre-Swan's theorem, one can do vector bundle theory on noncommutative manifold $(A,H,D)$, by replacing vector bundle by finitely generated projectve module $M$. For the construction of ...
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1 vote
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254 views

When splitting of short exact sequence preserves the kernels

This is a problem that I thought at first was obvious but that became less clear the more I thought about it. Assume we have a finitely generated algebra $A$ over a field $k$, and a short exact ...
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6 votes
1 answer
109 views

On the finiteness of an Auslander-Reiten component

I am reading a paper called A NOTE ON THE RADICAL OF A MODULE CATEGORY by CLAUDIA CHAIO AND SHIPING LIU. This is Theorem 2.7: And this is part of it's proof, in which the direction (2) $\Rightarrow $ ...
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infinite left degrees

I am reading a paper called A NOTE ON THE RADICAL OF A MODULE CATEGORY by CLAUDIA CHAIO AND SHIPING LIU. This is a part of the paper: Definition: Let $f: X \rightarrow Y$ be an irreducible morphism ...
1 vote
1 answer
124 views

About composition factors [closed]

I am reading a paper called A NOTE ON THE RADICAL OF A MODULE CATEGORY by CLAUDIA CHAIO AND SHIPING LIU. This is part of the proof of Lemma 2.3 $A$ is assumed to be an Artin algebra and mod(A) the ...
1 vote
1 answer
214 views

Question on simple modules and projective covers

I have the following question: Let $A$ be an Artin algebra. Let $S_1$ and $S_2$ be simple modules in $\text{mod}(A)$ and let $P(S_1)$ be the projective cover of $S_1$. Let $f: P(S_1) \rightarrow S_2$ ...
3 votes
2 answers
169 views

Question on injective hulls

How can I show the following: Let $f: M \rightarrow N$ be a morphism in $\text{mod}(A)$, where $A$ is an Artin algebra. Suppose $f \neq 0$. Then there exists a simple module $S$ with its injective ...
0 votes
1 answer
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injective hull and projective cover of simple modules are indecomposable

Let $A$ be an Artinian algebra. Let $S$ be a simple module over $A$. Let $\pi: S \rightarrow I$ be the injective hull and $\tau: P \rightarrow S$ be the projective cover of $S$. Then $I$ and $P$ must ...
6 votes
1 answer
118 views

Finitely presented modules admitting projective covers

A ring $R$ is called semi-perfect if every finitely generated $R$-module has a projective cover, and it can be proved that this is equivalent to say that the category consisting of the finitely ...
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0 votes
1 answer
395 views

Separability of $\mathbb{C}[x]$ over its $\mathbb{C}$-subalgebras

For commutative rings $R \subseteq S$, recall that $S$ is separable over $R$, if $S$ is a projective $S \otimes_R S$-module, via $f: S \otimes_R S \to S$ given by: $f(s_1 \otimes_R s_2)=s_1s_2$. ...
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5 votes
1 answer
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Trace ideal of a projective module

In his 1969 paper "On projective modules of finite rank", Wolmer Vasconcelos writes Let $M$ be a projective $R$-module... The trace of $M$ is defined to be the image of the map $M \otimes_R ...
5 votes
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Ring with different graded and ungraded global dimensions

Let $A$ be a $\mathbb N$-graded ring. One can consider the two categories $M_A^g$ and $M_A^u$ of graded and ungraded modules over $A$. Both have, say, enough projectives, hence one can compute various ...
1 vote
1 answer
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Second summand to make projective module free

Suppose there's a projective $R$-module $P$ (non-free). We know that there is another $R$-module $M$ such that $P\oplus M$ is free over $R$. Is there a way to write down such an $M$ in terms of $P$? ...
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1 answer
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Projective (or injective) object in a subcategory

Let $\mathcal{A}$ be an abelian category and $\mathcal{B}$ be a full subcategory of $\mathcal{A}$. Suppose that $\mathcal{B}$ is abelian and that the inclusion of $\mathcal{B}$ in $\mathcal{A}$ is ...
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5 votes
1 answer
144 views

Projective module which splits off sequence of submodules, but not the sum

Does there exist an example of a module $X$ over some ring $R$ together with submodules $T_i$ such that: $X$ is projective, $X$ splits as an internal direct sum $X\cong T_1\oplus T_2\oplus \ldots \...
2 votes
0 answers
47 views

Non-singular rings which are Rickart

A ring $R$ is said to be a right Rickart ring if the right annihilator of any element in $R$ is of the form $eR$ for some idempotent $e \in R$. It turns out that a ring $R$ is right Rickart iff every ...
11 votes
2 answers
1k views

Is every locally free module of rank $1$ over a commutative ring concretely invertible?

Since this subject is full of misunderstandings (see here, here, here, and here) let us fix a precise terminology. Let $A$ be a commutative ring and $P$ an $A$-module. I) We'll say that $P$ is a ...
9 votes
1 answer
447 views

Bézout ring with non-trivial Picard group?

[I asked this on stackexchange here a few weeks ago to no response] A ring is called Bézout when its finitely generated ideals are principal. Q: Is there a nice example of a Bézout ring $R$ with ...
2 votes
1 answer
261 views

Hom functor and arbitrary coproducts

Let $M$ be a finitely generated $R$-module. It's easy to check that, in this case, $\mathbf{Hom}(M,-)$ preserves infinite sums. Now suppose that $M$ is projective. Is the reciprocal true? That is, if ...
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8 votes
2 answers
512 views

Basis for free modules over an affine domain

Let $A=k[x_1,\cdots,x_r]/I$ for some prime ideal $I$ and some field $k$. Consider the free $A$-module $A^n$. Question 1. Given an element $e\in A^n$, is there a method to tell whether $e$ can be ...
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4 votes
0 answers
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Weyl theorem - possible corollary - alternative characterization of projective representation of $Z_N\times Z_N$

For an integer $N$, let $\omega=e^{2i\pi/N}$ and $A$, $B$ be the clock and shift operators: $A=\left(\begin{matrix} 1 & 0 & \cdots & 0 \\ 0 & \omega & \cdots & 0 \\ \vdots &...
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8 votes
1 answer
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Property of the trace on finitely generated projective modules

Let $A$ be a commutative ring with unit and let $P$ be a projective $A$-module finitely generated. By definition, there exists an $A$-module $P'$ such that $P\oplus P'$ is free of finite rank $r$. If $...
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2 votes
1 answer
132 views

A weak Schur's lemma for non-semisimple finite dimensional algebras

Let $B \subseteq C$ be an inclusion of finite dimensional (associative) algebras over a field $k$. Assume that $C$ is a free $B$-module. Let $\bigoplus_i U_i$ be a decomposition of $B$ into ...
1 vote
0 answers
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Lifting idempotents and projective coverings --- reference request

Let $R$ be a (non-commutative, unital) ring with Jacobson radical $J$, write $\overline{R}=R/J$ and denote the quotient map $R\to \overline{R}$ by $a\mapsto \overline{a}$. It is easy to see that if ...
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2 votes
1 answer
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A module associated to an endomorphism of a vector bundle

Let $E$ be a vector bundle over a compact connected Hausdorff space $X$. To an endomorphism $\alpha \in End(E)$, we associate a $C(X)-$ module $\Gamma(E,\alpha)$ consisting of all $\beta\in End(E)$ ...
5 votes
0 answers
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On existence of finitely generated projective generator with commutative endomorphism ring in ${}_R Mod$

Let $R$ be a ring with unity (not necessarily commutative). Let ${}_R Mod$ be the category of left $R$-modules. If ${}_R Mod$ has a projective generator with commutative endomorphism ring , then does $...
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6 votes
0 answers
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Projective modules over maximal orders of central simple algebras

In "Supersingular K3 surfaces" by TetsuJi Shioda, when proving Theorem 3.5 (Deligne) he considers a supersingular elliptic curve $C$ over an algebraic closed field of $\text{char}\ p>0$ and let $R ...
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7 votes
1 answer
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Moduli space of flat connections over a Riemann surface

If I understand correctly, in the Refs below: We can see that the moduli space of SU($N$) flat connections over a torus, is equivalent to a complex projective space $\mathbb{P}^{N-1}$ Namely, $$ M_{\...
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2 votes
0 answers
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Moduli space is a Calabi-Yau manifold?

I asked a question here where a moduli space of flat connection is related to the $n$-dimensional complex projective space: $$\Bbb E/S_n \cong \Bbb P^{n-1}. $$ This is related to a 4d SU(N) Yang-...
1 vote
1 answer
54 views

Decomposing semihereditary rings

Let $R$ be a commutative semihereditary ring (i.e. every finitely generated ideal of $R$ is projective https://en.wikipedia.org/wiki/Hereditary_ring). Then is $R$ a finite direct product of Prufer ...
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1 vote
1 answer
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Decomposing Noetherian hereditary rings of Krull dimension $1$ into product of hereditary domains (i.e. Dedekind domains)

Let $R$ be a commutative Noetherian hereditary ring (https://en.wikipedia.org/wiki/Hereditary_ring) of Krull dimension $1$. Then is it true that $R$ is a finite direct product of Dedekind domains ?
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1 vote
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Numerator-cancellable Modules

I don't know why there is no investigation of the cancellability of quotients in category of modules. What I mean by cancellability of quotients in category of modules is the following : Let $R$ be a ...
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5 votes
1 answer
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Are these two constructions of $K_0(A)$ isomorphic?

The following question is extracted from this question on MSE, which got no answer so far, probably because it was a bit hidden by another question which a posteriori was totally obvious. Let $A$ be ...
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1 vote
0 answers
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structure of flat modules over a DVR

I. Kaplansky has proved that if a torsion free module (over a complete DVR) is countably generated and does not contains infinitely divisible elements then it is free. Is there any analog of this ...